PR V MA3041 GEOMETRI

KEVIN MANDIRA LIMANTA

(10110106)1. Prove that lines P + [v] and Q+ [w] intersect if and only ifQP, v w = 0Answer:Let P + [v] and Q+ [w] intersect at R, then we haveR = P +v = Q+wso that QP = v w. Hence, we haveQP, v w = v w, v w= v, v w w, v w= 0as desired.Conversely, suppose that Q P, v w = 0, then we can represent Q P as linear combinationof v and w, namelyQP = v +wHence we have that P + v = Q w. But the left side is on line P + [v], while the right side ison line Q+ [w]. Since both sides are equal, it must be the case that those two lines intersect.2. Let A and B be two distinct points of S2. Show thatX S2[ d(X, A) = d(X, B)is a line, and nd an expression for its pole.Answer:To show that l = X S2[ d(X, A) = d(X, B) is a line, we must show that there existsvector , which is its pole, such that X, = 0 for all X l. But sinced(X, A) = cos1X, A = cos1X, B = d(X, B),we have thatX, A = X, Bor X, A B = 0.Hence, we have shown the existence of , namely AB, and l is a line X S2[ X, AB = 0.13. Prove Theorem 11. In particular, show that(a) The poles of m are P[ P[, where is a pole of l.(b) The point of intersection are P P, _1 P, 2.(c) The distances from P to l are cos1_

_1 P, 2_Answer:Choose m =_X S2[_X,

OP [OP [_= 0_. To prove the uniqueness, suppose that there exists such that , = 0 and the line that has as its pole also passes through P. Then we have and P is on the line, and the line connecting any two distinct points must be unique.(a) Since the line m that is perpendicular to l is unique, then we must have = OP [OP [(b) Vector that is related is the projection of OP towards , that is P P, . We must have theunit vector, so the point of intersection is

3But every element in above set is an inverse of itself except for HPHQ. It must be the case thenthat HPHQ is the inverse of itself, that is if they commute. Hence we must have HPHQ = HQHP.Direct calculation shows thatHPHQ(X) = X 2X, QQ 2X, PP + 4X, QP, QPwhileHQHP(X) = X 2X, QQ 2X, PP + 4X, PP, QQSo, we must haveX, QP, QP = X, PP, QQLeft side is a multiple of P, while right side is a multiple of Q. So they are equal if and only ifP, Q = 0, that is when vectors OP and OQ are orthogonal.6. If an isometry of S2leaves P xed and takes Q to Q, show that P, Q = 0.Answer:Suppose T is the isometry, then we have TP = P and TQ = Q. We then haved(P, Q) = d(TP, TQ) = d(P, Q)so thatcos1P, Q = cos1P, Qor P, Q = P, Q = P, Q. This gives P, Q = 0, as desired.7. Find all isometries of S2satisfying 2= I, but ,= I. Such an isometry is said to be an involu-tion. If and are involutions, is an involution?Answer:Reection, rotation about , half-turn, and glide reection, which is reection followed by transla-tion about . Product of involutions and need not be involution. Take and such that theydont commute, that is, ,= . There exists such a pair, take = l and = m where l andm are not perpendicular.8. Let P be a point of S2, and l a line of S2. Show that(lHP)2= Iif and only if P is a pole of l or P l.Answer:() We divide this into two cases. Let P be a pole of l. For any x S2, we haveHPx = x + 2x, P and l = x 2x, P4Consequently, we have

lP5We have three cases:(a) If y, P = 0, then P is pole of l(b) If P, = 0, then P is in l(c) If lP = 0, then |lP| = 0, which contradicts the fact that |lP| = 1 because lP S2.In every cases, P is pole of l or P l.9. Let l be a line of S2with pole P. Show that I, HP, l, E is a group, and give its multiplicationtable. (E is the antipodal map)Answer:Since associative property holds and there is I as identity element, it suces to show that ev-ery element has an inverse. Clearly I is the inverse of itself, as well as HP and l because they areall involutions. Note that we have EE = lHP

lHP = l

lHPHP = I, so that E is an inverse ofitself also.The multiplication table is given as follows: I HP

l EI I HP

l EHP HP I E l

l

l E I HPE E l HP I10. Let F = P, Q, R be a gure consisting of three mutually perpendicular points. Find S(F).Answer:Because P, Q, and R are three mutually perpendicular points, then there are lines in S2whichpoles are P, Q, and R. Then the group S(F) consists of 24 rotations, 9 reections, and 15 glidereections, including the antipodal map.11. Under what circumstances will a reection and a half-turn commute?Answer:Let lx = x 2x, QQ with Q pole of l, and HPx = x + 2x, PP. We have